(8)
These definitions require some elucidation. The object of
Geometry1
is the properties of figure, and figure is defined to be
the relation which subsists between the boundaries of space.
Space or magnitude is of three kinds, line,
surface, and solid. It may be here observed,
once for all, that the terms used in geometrical science, are not
designed to signify any real, material or physical existences.
They signify certain abstracted notions or conceptions of the
mind, derived, without doubt, originally from material objects by
the senses, but subsequently corrected, modified, and, as it
were, purified by the operations of the understanding. Thus, it
is certain, that nothing exactly conformable to the geometrical
notion of a right line ever existed; no edge, which the finest
tool of an artist can construct, is so completely free from
inequalities as to entitle it to be consisdered as a mathematical
right line. Nevertheless, the first notion of such an edge being
obtained by the senses, the process of mind by which we reject
the inequalities incident upon the nicest mechanical production,
and substitute for them, mentally, that perfect evenness
which constitutes the essence of a right line, is by no means
difficult. In like manner, if a pen be drawn over this paper an
effect is produced, which, in common language, would be called a
line, right or curved, as the case may be. This, however,
cannot, in the strict geometrical sense of the term, be a
line at all, since it has breadth as well as length; for
if it had not it could not be made evident to the senses. But
having first obtained this rude and incorrect notion of a line,
we can imagine that, while its length remains unaltered, it may
be infinitely attenuated until it ceases alteogether to have
breadth, and thus we obtain the exact conception of a
mathematical line.

The different modes of magnitude are ideas so extremely
uncompounded that their names do not admit of definition properly
so called at
all.2
We may, however, assist the student to form correct notions of
the true meaning of these terms, although we may not give
rigorous logical definitions of them.

A notion being obtained by the senses of the smallest magnitude
distinctly perceptible, this is called a physical point.
If this point were indivisible even in idea, it would be
strictly what is called a mathematical point. But this
is not the case. No material substance can assume a magnitude so
small that a smaller may not be imagined. The mind, however,
having obtained the notion of an extremely minute magnitude, may
proceed without limit in a mental diminution of it; and that
state at which it would arrive if this diminution were infinitely
continued is a mathematical
point.3

The introduction of the idea of motion into geometry has been
objected to as being foreign to that science. Nevertheless, it
seems very doubtful whether we may not derive from motion the
most distinct ideas of the modes of magnitude. If a mathematical
point be conceived to move in space, and to mark its course by a
trace or track, that trace or track will be a mathematical
line. As the moving point has no magnitude, so it is
evident that its track can have no breadth or thickness. The
places of the point at the beginning and end of its motion, are
the extremities of the line, which are therefore points.
The third of the preceding definitions is not properly a
definition, but a proposition, the truth of which may be inferred
from the first two definitions.

As a mathematical line may be conceived to proceed from
the motion of a mathematical point, so a
physical line may be conceived to be generated by the
motion of a physical point.

In the same manner as the motion of a point determines the idea
of a line, the motion of a line may give the idea of a surface.
If a mathematical line be conceived to move, and to leave in the
space through which it passes a trace or track, this trace or
track will be a surface; and since the line has no breadth, the
surface can have no thickness. The initial and final positions
of the moving line are two boundaries or extremities of the
surface, and the other extremities are the lines traced by the
extreme points of the line whose motion produced the surface.

The sixth definition is therefore liable to the same objection as
the third. It is not properly a definition, but a principle, the
truth of which be derived from the fifth and preceding
definitions.

It is scarcely necessary to observe, that the validity of the
objection against introducing motion as a
principle into the Elements of Geometry, is not here
disputed, nor is it introduced as such. The preceding
observations are designed merely as illustrations to
assist the student in forming correct notions of the true
mathematical significations of the different modes of magnitude.
With the same view we shall continue to refer to the same
mechanical ideas of motion, and desire our observations always to
be understood in the same sense.

The fourth definition, that of a right or straight line, is
objectionable, as being unintelligible; and the same may be said
of the definition (seventh) of a plane surface. Those who do not
know what the words ‘straight line’ and ‘plane
surface’ mean, will never collect their meaning from these
definitions; and to those who do know the meaning of
those terms, definitions are useless. The meaning of the terms
‘right line’ and ‘plane surface’ are only
to be made known by an appeal to experience, and the evidence of
the senses, assisted, as was before observed, by the power of the
mind called abstraction. If a perfectly flexible string
be pulled by its extremities in opposite directions, it will
assume, between the two points of tension, a certain position.
Were we to speak without the rigorous exactitude of geometry, we
should say that it formed a straight line. But upon
consideration, it is plain that the string has weight, and that
its weight produces a flexure in it, the convexity of which will
be turned towards the surface of the earth. If we conceive the
weight of the string to be extremely small, that flexure will be
proportionably small, and if, by the process of abstraction, we
conceive the string to have no weight, the flexure will
altogether disappear, and the string will be accurately a
straight line.

A straight line is sometimes defined ‘to be the shortest
way between two points.’ This is the definition given by
Archimedes, and after him by Legendre in his Geometry; but Euclid
considers this as a property to be proved. In this sense, a
straight line may be conceived to be that which is traced by one
point moving towards another, which is quiescent.

Plato defines a straight line to be that whose extremity hides
all the rest, the eye being placed in the continuation of the
line.

Probably the best definition of a plane surface is, that it is
such a surface that the right line, which joins every two points
which can be assumed upon it, lies entirely in the surface. This
definition, originally given by Hero, is substituted for
Euclid's by R. Simson and Legendre.

Plato defined a plane surface to be one whose extremities hide
all the intermediate parts, the eye being placed in its
continuation.

It has been also defined as ‘the smallest surface which can
be contained between given extremities.’

Every line which is not a straight line, or composed of straight
lines, is called a curve. Every surface which is not a
plane, or composed of planes, is called a curved
surface.

(14)
Angles might not improperly be considered as a fourth species of
magnitude. Angular magnitude evidently consists of parts, and
must therefore be admitted to be a species of quantity. The
student must not suppose that the magnitude of an angle is
affeced by the length of the right lines which include it, and of
whose mutual divergence it is the measure. These lines, which
are called the sides or legs of the angle, are
supposed to be of indefinite length. To illustrate the nature of
angular magnitude, we shall again recur to motion.
CAA0A1A2A3A4A5A6
Let
C
be supposed to be the extremity of a right line
C A,
extending indefinitely in the direction
C A.
Through the same point
C,
let another indefinite straight line
C A0,
be conceived to be drawn; and suppose
this right line to revolve in the same plane round
its extremity
C,
it being supposed at the beginning of
its motion to coincide with
C A.
As it revolves from
C A0
to
C A1,
C A2,
C A3,
&c., its divergence from
C A
or, what is the same, the angle it makes with
C A,
continually increases. The line continuing to revolve,
and successively assuming the positions
C A1,
C A2,
C A3,
C A4,
&c., will at length coincide with the continuation
C A5
of the line
C A0
on the opposite side of the
point
C.
When it assumes this position, it is considered
by Euclid to have no inclination to
C A0,
and to form no angle with it. Nevertheless, when
the student advances further in mathematical science, he
will find, that not only the line
C A5
is considered
to form an angle with
C A0,
but even when the revolving line continues its motion past
C A6;
and this angle is measured in the direction
A6,
A5,
A4,
&c. to
A0.

The point where the sides of an angle meet is called the
vertex of the angle.

Superposition is the process by which one magnitude may
be conceived to be placed upon another, so as exactly to cover
it, or so that every part of each shall exactly coincide with
every part of the other.

It is evident that any magnitudes which admit of superposition
must be equal, or rather this may be considered as the definition
of equality. Two angles are therefore equal when they admit of
superposition.
AA′BB′CC′
This may be determined thus; if the angles
A B C
and
A′ B′ C′
are those whose equality is to be ascertained, let the vertex
B′
be conceived to be placed on the vertex
B,
and the side
B′ A′
on the side
B A,
and let the remaining side
B′ C′
be placed on the same side of
B A
with
B C.
If under these circumstances
B′ C′
lie upon, or coincide with
B C,
the angles admit of superposition, and are equal, but
are otherwise not. If the side
B′ C′
fall between
B C
and
B A,
the angle
B′
is said to be less than the angle
B,
and if the side
B C
fall between
B′ C′
and
B A,
the angle
B′
is said to be greater than
B.

As soon as the revolving line assumes such a position
C A3
that the angle
A C A3
is equal to the angle
A3 C A5,
each of those angles is called a right angle.

An angle is sometimes expressed simply by the letter placed at
its vertex, as we have done in comparing the angles
B
and
B′.
But when the same point, as
C,
is the vertex of more angles than one, it is necessary
to use the three letters expressing the
sides as
A C A3,
A3 C A5,
the letter at the vertex being always placed in the middle.

When a line is extended, prolonged, or has its length increased,
it is said to be produced, and the increase of length
which it receives is called its produced part, or its
production.
ABB′
Thus, if the right line
A B
be prolonged to
B′,
it is said to be produced through the
extremity
B,
and
B B′
is called its production or produced part.

Two lines which meet and cross each other are said to
intersect, and the point or points where they meet are
called points of intersection. It is assumed as a
self-evident truth, that two right lines can only intersect in
one point. Curves, however, may intersect each other, or right
lines, in several points.

Two right lines which intersect, or whose productions intersect,
are said to be inclined to each other, and their
inclination is measured by the angle which they include. The
angle included by two right lines is sometimes called the angle
under those lines; and right lines which include equal
angles are said to be equally inclined to each other.

It may be observed, that in general when right lines and plane
surfaces are spoken of in Geometry, there are considered as
extended or produced indefinitely. When a determinate
portion of a right line is spoken of, it is generally called a
finite right line. When a right line is said to be
given, it is generally meant that its position or direction on a
plane is given. But when a finite right line is given,
it is understood, that not only its position, but its length is
given. These distinctions are not always rigorously observed,
but it never happens that any difficulty arises, as the meaning
of the words is always sufficiently plain from the context.

When the direction alone of a line is given, the line is
sometimes said to be given in position, and when the
length alone is given, it is said to be given in
magnitude.

By the inclination of two finite right lines which do not meet,
is meant the angle which would be contained under these lines if
produced until they intersect.

A circle is a plane figure, bounded by one
continued line, called its circumference or
periphery; and having a certain point within it, from
which all right lines drawn to its circumference are equal.

If a right line of a given length revolve in the same plane round
one of its extremities as a fixed point, the other extremity will
describe the circumference of a circle, of which the centre is
the fixed extremity.

A semicircle is the figure contained by the diameter, and
the part of the circle cut off by the diameter.

(22)
From the definition of a circle, it follows immediately, that a
line drawn from the centre to any point within the
circle is less than the radius; and a line from the centre to any
point without the circle is greater than the radius.
Also, every point, whose distance from the centre is
less than the radius, must be within the
circle; every point whose distance from the centre is
equal to the radius must be on the circle;
and every point, whose distance from the centre is
greater than the radius, is without the circle.

The word ‘semicircle’ in Def. XVIII., assumes, that a
diameter divides the circle into two equal parts. This may be
easily proved by supposing the two parts, into which the circle
is thus divided, placed one upon the other, so that they shall
lie at the same side of their common diameter: then if the arcs
of the circle which bound them do not coincide, let a radius be
supposed to be drawn, intersecting them. Thus, the radius of the
one will be a part of the radius of the other; and therefore, two
radii of the same circle are unequal, which is contrary to the
definition of a circle (17.)

A triangle is the most simple of all rectilinear figures, since
less than three right lines cannot form any figure. All other
rectilinear figures may be resolved into triangles by drawing
right lines from any point within them to their several vertices.
The triangle is therefore, in effect, the element of all
rectilinear figures; and on its properties, the properties of all
other rectilinear figures depend. Accordingly the greater part
of the first book is devoted to the development of the properties
of this figure.

Polygons are called pentagons, hexagons, heptagons, &c.,
according as they are bounded by five, six, seven or more sides.
A line joining the vertices of any two angles which are not
adjacent is called a diagonal of the polygon.

In general, all rectilinear figures whose sides are equal, may be
said to be equilateral.

Two rectilinear figures, whose sides are respectively equal each
to each, are said to be mutually equilateral. Thus, if
two triangles have each sides of three, four, or five feet in
length, they are mutually equilateral, although neither
of them is an equilateral triangle.

In the same way a rectilinear figure having all its angles equal,
is said to be equiangular, and two rectilinear figures
whose several angles are equal each to each, are said to be
mutually equiangular.

We have ventured to change the definition of a square as given in
the text. A lozenge, called by Euclid a rhombus, when
equiangular, must have all its angles right, as will appear
hereafter. Euclid's definition, which is a ‘a lozenge all
whose angles are right,’ therefore, contains more than
sufficient for a definition, inasmuch as, had the angles been
merely defined to be equal, they might be
proved to be right. To effect this change in the
definition of a square, we have transposed the order of the last
two definitions. See (158).

An oblong is a quadrilateral, whose angles are all
right, but whose sides are not equal.

This term is not used in the Elements, and therefore the
definition might have been omitted. The same figure is defined
in the second book, and called a rectangle. It would
appear that this circumstance of defining the same figure twice
must be an oversight.

This definition and the term rhomboid are superceded by
the term parallelogram, which is a quadrilateral, whose
opposite sides are parallel. It will be proved hereafter, that
if the opposite sides of a quadrilateral be equal, it must be a
parallelogram. Hence, a distinct denomination for such a figure
is useless.

Parallel right lines are such as are in the same plane, and
which, being produced continually in both direction,
would never meet.

It should be observed, that the circumstance of two right lines,
which are produced indefinitely, never meeting, is not sufficient
to establish their parallelism. For two right lines which are
not in the same plane can never meet, and yet are not parallel.
Two things are indispensably necessary to establish the
parallelism of two right lines, 1°, that they be in
the same plane, and 2°, that when indefinitely produced, they
never meet. As in the first six books of the Elements all the
lines which are considered are supposed to be in the same plane,
it will be only necessary to attend to the latter criterion of
parallelism.

Notes on the Definitions

2
The name of a simple idea cannot be defined, because the general
terms which compose the definition signifying several different
ideas can by no means express an idea which has no manner of
composition.—Locke.